Dear @Christian, I guess it depends what you want to do. For example, does the spectral theorem give you a bound on how many discrete eigenvalues are to be found in a given bounded interval ? I number theory we have the so-called Selberg Trace formula that provides such a bound and in order to get it one needs to get a good hold on these "generalized eigenvectors".

Well I like when I can touch and compute things. For example Eisenstein series admit Fourier series expansion and this gives you a good hold on these objects. I read a little bit about the spectral theorem for unbounded self-adjoint operators and I did not get much insight regarding the associated spectral measure. At the end I would like to be able for any f in H to write it as a weighted sum of explicit eigenvectors for which I have a good hold.

Dear @ChristianRemling, I agree that my expression "physical incarnation" is vague. The best examples that come to my mind are the functions $x\mapsto e^{ix\xi}$ where $\xi\in \mathbb{R}$ for the usual Laplacian $-d/dx^2$ on $\mathbb{R}$. These function are not square integrable but they are still "small" since their absolute value is equal to 1 as $x$ varies over $\mathbb{R}$. Other simple examples are Eisenstein series when one replaces $\mathbb{R}$ by the Poincare upper half-plane quotiented by a Fuchsian group of the first kind.